Asymptotics for the spectral function on Zoll manifolds

Let $(M,g)$ be a Zoll manifold, i.e., a smooth, compact, Riemannian manifold without boundary all of whose geodesics are closed with a minimal common period $T$. The positive definite Laplace-Beltrami operator has eigenvalues $\{{\lambda }_j^2{\}}_j$ which cluster around ${\nu }_{\mathrm{\ell }}^2$ for some sequence ${\nu }_{\mathrm{\ell }}\to \mathrm{\infty }$. This article is concerned with the number of ${\lambda }_j$ in a window of fixed size $\mathrm{w}$ around ${\nu }_{\mathrm{\ell }}$, denoted by $\mathbf{N}({\nu }_{\mathrm{\ell }},\mathrm{w}):=\mathrm{\#}\{j:{\lambda }_j\in [{\nu }_{\mathrm{\ell }}-\mathrm{w},{\nu }_{\mathrm{\ell }}+\mathrm{w}]\}\mathrm{.}$ When the set of trajectories with period smaller than $T$ has zero measure, there is $c_n>0$, depending only on $n=\dim M$, such that $$\mathbf{N}({\nu }_{\mathrm{\ell }},\mathrm{w})=c_n{\mathrm{vol}}_g(M){\nu }_{\mathrm{\ell }}^{n-1}+o({\nu }_{\mathrm{\ell }}^{n-1}),$$ as $\mathrm{\ell }\to \mathrm{\infty }$. However, for a general Zoll manifold this may not be the case. We show that, nevertheless, there is $N>0$, independent of $\mathrm{\ell }$, such that $$\sum _{j=0}^{N-1}\mathbf{N}({\nu }_{\mathrm{\ell }+j},\mathrm{w})=c_{n}N{\mathrm{vol}}_g(M){\nu }_{\mathrm{\ell }}^{n-1}+o({\nu }_{\mathrm{\ell }}^{n-1}),$$ as $\mathrm{\ell }\to \mathrm{\infty }$. In addition to asymptotics for the counting function, we study the kernel of the spectral projector for the Laplacian, ${\mathrm{\Pi }}_{\mathrm{\ell },\mathrm{w}}(x,y)$ onto the spectrum in ${\bigcup }_{j=0}^{N-1}[{\nu }_{\mathrm{\ell }+j}-\mathrm{w},{\nu }_{\mathrm{\ell }+j}+\mathrm{w}]$. We show that for $x$ and $y$ in a shrinking neighborhood of a point with few loops of length smaller than $T$, ${\mathrm{\Pi }}_{\mathrm{\ell },\mathrm{w}}(x,y)$ and its derivatives have the same asymptotics as those on the round sphere and flat torus.